Function spaces | Sobolev spaces | Fourier analysis | Fractional calculus

Sobolev space

In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function. Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that weak solutions of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces of continuous functions with the derivatives understood in the classical sense. (Wikipedia).

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Related pages

Norm (mathematics) | Differential equation | Vector space | Separable space | Normed vector space | Sobolev inequality | Derivative | Almost everywhere | Fourier series | Continuous function | Sobolev space | Besov space | Hölder condition | Banach space | Algebra over a field | Partial derivative | Lebesgue integration | Unit ball | Parseval's theorem | Poincaré inequality | Natural number | Mathematics | Weak derivative | Distribution (mathematics) | Interpolation space | Embedding | Lipschitz continuity | Orthonormal basis | Cocompact embedding | Trace operator | Cantor function | Compact operator | Dirichlet boundary condition | Uniform norm | Hilbert space | Weak solution | Laplace operator | Absolute continuity | Sobolev mapping | Integration by parts | Lp space | Partial differential equation | Inner product space | Cone condition | Domain (mathematical analysis) | Meyers–Serrin theorem | Complete metric space | Dirichlet problem | Multiplier (Fourier analysis)